Gr\"obner bases for (all) Grassmann manifolds

TL;DR

This paper develops Gr"obner bases for the ideals defining the mod 2 cohomology of Grassmann manifolds, enabling new immersion results for an infinite family of these manifolds.

Contribution

It determines reduced Gr"obner bases for the cohomology ideals of Grassmann manifolds and applies them to prove new immersion theorems.

Findings

Determined reduced Gr"obner bases for cohomology ideals of Grassmann manifolds.

Proved new immersion theorems for an infinite family of Grassmann manifolds.

Established a connection between algebraic computations and geometric topology results.

Abstract

Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to understand this cohomology via Gr\"obner bases. Reduced Gr\"obner bases for the ideals $I_{k,n}$ are determined. An application of these bases is given by proving an immersion theorem for Grassmann manifolds $G_{5,n}$, which establishes new immersions for an infinite family of these manifolds.